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<div class="titlepage"><div><div><h4 class="title">
<a name="math_toolkit.dist_ref.dists.exp_dist"></a><a class="link" href="exp_dist.html" title="Exponential Distribution">Exponential Distribution</a>
</h4></div></div></div>
<pre class="programlisting"><span class="preprocessor">#include</span> <span class="special">&lt;</span><span class="identifier">boost</span><span class="special">/</span><span class="identifier">math</span><span class="special">/</span><span class="identifier">distributions</span><span class="special">/</span><span class="identifier">exponential</span><span class="special">.</span><span class="identifier">hpp</span><span class="special">&gt;</span></pre>
<pre class="programlisting"><span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span> <span class="special">=</span> <span class="keyword">double</span><span class="special">,</span>
          <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a>   <span class="special">=</span> <a class="link" href="../../pol_ref/pol_ref_ref.html" title="Policy Class Reference">policies::policy&lt;&gt;</a> <span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">exponential_distribution</span><span class="special">;</span>

<span class="keyword">typedef</span> <span class="identifier">exponential_distribution</span><span class="special">&lt;&gt;</span> <span class="identifier">exponential</span><span class="special">;</span>

<span class="keyword">template</span> <span class="special">&lt;</span><span class="keyword">class</span> <span class="identifier">RealType</span><span class="special">,</span> <span class="keyword">class</span> <a class="link" href="../../../policy.html" title="Chapter 22. Policies: Controlling Precision, Error Handling etc">Policy</a><span class="special">&gt;</span>
<span class="keyword">class</span> <span class="identifier">exponential_distribution</span>
<span class="special">{</span>
<span class="keyword">public</span><span class="special">:</span>
   <span class="keyword">typedef</span> <span class="identifier">RealType</span> <span class="identifier">value_type</span><span class="special">;</span>
   <span class="keyword">typedef</span> <span class="identifier">Policy</span>   <span class="identifier">policy_type</span><span class="special">;</span>

   <span class="identifier">exponential_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">lambda</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>

   <span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
<span class="special">};</span>
</pre>
<p>
          The <a href="http://en.wikipedia.org/wiki/Exponential_distribution" target="_top">exponential
          distribution</a> is a <a href="http://en.wikipedia.org/wiki/Probability_distribution" target="_top">continuous
          probability distribution</a> with PDF:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../equations/exponential_dist_ref1.svg"></span>

          </p></blockquote></div>
<p>
          It is often used to model the time between independent events that happen
          at a constant average rate.
        </p>
<p>
          The following graph shows how the distribution changes for different values
          of the rate parameter lambda:
        </p>
<div class="blockquote"><blockquote class="blockquote"><p>
            <span class="inlinemediaobject"><img src="../../../../graphs/exponential_pdf.svg" align="middle"></span>

          </p></blockquote></div>
<h5>
<a name="math_toolkit.dist_ref.dists.exp_dist.h0"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.exp_dist.member_functions"></a></span><a class="link" href="exp_dist.html#math_toolkit.dist_ref.dists.exp_dist.member_functions">Member
          Functions</a>
        </h5>
<pre class="programlisting"><span class="identifier">exponential_distribution</span><span class="special">(</span><span class="identifier">RealType</span> <span class="identifier">lambda</span> <span class="special">=</span> <span class="number">1</span><span class="special">);</span>
</pre>
<p>
          Constructs an <a href="http://en.wikipedia.org/wiki/Exponential_distribution" target="_top">Exponential
          distribution</a> with parameter <span class="emphasis"><em>lambda</em></span>. Lambda
          is defined as the reciprocal of the scale parameter.
        </p>
<p>
          Requires lambda &gt; 0, otherwise calls <a class="link" href="../../error_handling.html#math_toolkit.error_handling.domain_error">domain_error</a>.
        </p>
<pre class="programlisting"><span class="identifier">RealType</span> <span class="identifier">lambda</span><span class="special">()</span><span class="keyword">const</span><span class="special">;</span>
</pre>
<p>
          Accessor function returns the lambda parameter of the distribution.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.exp_dist.h1"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.exp_dist.non_member_accessors"></a></span><a class="link" href="exp_dist.html#math_toolkit.dist_ref.dists.exp_dist.non_member_accessors">Non-member
          Accessors</a>
        </h5>
<p>
          All the <a class="link" href="../nmp.html" title="Non-Member Properties">usual non-member accessor
          functions</a> that are generic to all distributions are supported:
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.cdf">Cumulative Distribution Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.pdf">Probability Density Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.quantile">Quantile</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.hazard">Hazard Function</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.chf">Cumulative Hazard Function</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mean">mean</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.median">median</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.mode">mode</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.variance">variance</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.sd">standard deviation</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.skewness">skewness</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis">kurtosis</a>, <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.kurtosis_excess">kurtosis_excess</a>,
          <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.range">range</a> and <a class="link" href="../nmp.html#math_toolkit.dist_ref.nmp.support">support</a>.
        </p>
<p>
          The domain of the random variable is [0, +∞].
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.exp_dist.h2"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.exp_dist.accuracy"></a></span><a class="link" href="exp_dist.html#math_toolkit.dist_ref.dists.exp_dist.accuracy">Accuracy</a>
        </h5>
<p>
          The exponential distribution is implemented in terms of the standard library
          functions <code class="computeroutput"><span class="identifier">exp</span></code>, <code class="computeroutput"><span class="identifier">log</span></code>, <code class="computeroutput"><span class="identifier">log1p</span></code>
          and <code class="computeroutput"><span class="identifier">expm1</span></code> and as such should
          have very low error rates.
        </p>
<h5>
<a name="math_toolkit.dist_ref.dists.exp_dist.h3"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.exp_dist.implementation"></a></span><a class="link" href="exp_dist.html#math_toolkit.dist_ref.dists.exp_dist.implementation">Implementation</a>
        </h5>
<p>
          In the following table λ is the parameter lambda of the distribution, <span class="emphasis"><em>x</em></span>
          is the random variate, <span class="emphasis"><em>p</em></span> is the probability and <span class="emphasis"><em>q
          = 1-p</em></span>.
        </p>
<div class="informaltable"><table class="table">
<colgroup>
<col>
<col>
</colgroup>
<thead><tr>
<th>
                  <p>
                    Function
                  </p>
                </th>
<th>
                  <p>
                    Implementation Notes
                  </p>
                </th>
</tr></thead>
<tbody>
<tr>
<td>
                  <p>
                    pdf
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: pdf = λ * exp(-λ * x)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    cdf
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: p = 1 - exp(-x * λ) = -expm1(-x * λ)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    cdf complement
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: q = exp(-x * λ)
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: x = -log(1-p) / λ = -log1p(-p) / λ
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    quantile from the complement
                  </p>
                </td>
<td>
                  <p>
                    Using the relation: x = -log(q) / λ
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mean
                  </p>
                </td>
<td>
                  <p>
                    1/λ
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    standard deviation
                  </p>
                </td>
<td>
                  <p>
                    1/λ
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    mode
                  </p>
                </td>
<td>
                  <p>
                    0
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    skewness
                  </p>
                </td>
<td>
                  <p>
                    2
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    kurtosis
                  </p>
                </td>
<td>
                  <p>
                    9
                  </p>
                </td>
</tr>
<tr>
<td>
                  <p>
                    kurtosis excess
                  </p>
                </td>
<td>
                  <p>
                    6
                  </p>
                </td>
</tr>
</tbody>
</table></div>
<h5>
<a name="math_toolkit.dist_ref.dists.exp_dist.h4"></a>
          <span class="phrase"><a name="math_toolkit.dist_ref.dists.exp_dist.references"></a></span><a class="link" href="exp_dist.html#math_toolkit.dist_ref.dists.exp_dist.references">references</a>
        </h5>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; ">
<li class="listitem">
              <a href="http://mathworld.wolfram.com/ExponentialDistribution.html" target="_top">Weisstein,
              Eric W. "Exponential Distribution." From MathWorld--A Wolfram
              Web Resource</a>
            </li>
<li class="listitem">
              <a href="http://documents.wolfram.com/calccenter/Functions/ListsMatrices/Statistics/ExponentialDistribution.html" target="_top">Wolfram
              Mathematica calculator</a>
            </li>
<li class="listitem">
              <a href="http://www.itl.nist.gov/div898/handbook/eda/section3/eda3667.htm" target="_top">NIST
              Exploratory Data Analysis</a>
            </li>
<li class="listitem">
              <a href="http://en.wikipedia.org/wiki/Exponential_distribution" target="_top">Wikipedia
              Exponential distribution</a>
            </li>
</ul></div>
<p>
          (See also the reference documentation for the related <a class="link" href="extreme_dist.html" title="Extreme Value Distribution">Extreme
          Distributions</a>.)
        </p>
<div class="itemizedlist"><ul class="itemizedlist" style="list-style-type: disc; "><li class="listitem">
              <a href="https://www.google.com/books/edition/Extreme_Value_Distributions/GwBqDQAAQBAJ?hl=en&amp;gbpv=0" target="_top">Extreme
              Value Distributions, Theory and Applications Samuel Kotz &amp; Saralees
              Nadarajah</a> discuss the relationship of the types of extreme
              value distributions.
            </li></ul></div>
</div>
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      Gautam Sewani, Benjamin Sobotta, Nicholas Thompson, Thijs van den Berg, Daryle
      Walker and Xiaogang Zhang<p>
        Distributed under the Boost Software License, Version 1.0. (See accompanying
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